Ore Extensions Research Articles

Armendariz modules over skew PBW extensions

The aim of this article is to develop the theory of skew Armendariz and quasi-Armendariz modules over skew PBW extensions. We generalize the results of several works in the literature concerning these topics for the context of Ore extensions to another non-commutative rings which can not be expressed as iterated Ore extensions. As a consequence of our treatment, we extend and unify different results about the Armendariz, Baer, p.p., and p.q.-Baer properties for Ore extensions and skew PBW extensions.

Elementary Matrix-computational Proof of Quillen-Suslin Theorem for Ore Extensions

Elementary Matrix-computational Proof of Quillen-Suslin Theorem for Ore Extensions

Artin-Schelter regularity of twisted tensor products

We prove an Artin-Schelter regularity result for the method of twisted tensor products under a certain form. Such twisted tensor products, whose twisting maps are determined by the action on the generators, include Ore extensions and double Ore extensions. It is helpful to construct high-dimensional Artin-Schelter regular algebras.

Skew Poincaré–Birkhoff–Witt extensions over weak zip rings

In this paper we study skew Poincare–Birkhoff–Witt extensions over weak zip rings. Since these extensions generalize Ore extensions of injective type and another noncommutative rings of polynomial type, we unify and extend several results in the literature concerning the zip property. Under adequate conditions, we transfer this property from a ring of coefficients to a skew Poincare–Birkhoff–Witt extension over this ring. We illustrate our results with examples of noncommutative algebras appearing in noncommutative algebraic geometry and theoretical physics.

HOM-ASSOCIATIVE ORE EXTENSIONS AND WEAK UNITALIZATIONS

We introduce hom-associative Ore extensions as non-unital, non-associative Ore extensions with a hom-associative multiplication, and give some necessary and sufficient conditions when such exist. Within this framework, we construct families of hom-associative quantum planes, universal enveloping algebras of a Lie algebra, and Weyl algebras, all being hom-associative generalizations of their classical counterparts, as well as prove that the latter are simple. We also provide a way of embedding any multiplicative hom-associative algebra into a multiplicative, weakly unital hom-associative algebra, which we call a weak unitalization.

Algunas propiedades homológicas del plano de Jordan

El plano de Jordan puede ser visto como un álgebra cociente, como una extensión de Ore graduada y como unaextensión PBW torcida graduada. Usando estas interpretaciones, se muestra de forma explícita que el plano de Jordan es un álgebra Artin-Schelter regular y Calabi-Yau torcida, además se calcula de forma explícita su automorfismo de Nakayama.

Realization of PBW-deformations of type [formula omitted] quantum groups via multiple Ore extensions

The notion of multiple Ore extension is introduced as a natural generalization of Ore extensions and double Ore extensions. For a PBW-deformation Bq(sl(n+1,C)) of type An quantum group, we explicitly obtain the commutation relations of its root vectors, then show that it can be realized via a series of multiple Ore extensions, which we call a ladder Ore extension of type (1,2,⋯,n). Moreover, we analyze the quantum algebras Bq(g) with g of type D4, B2 and G2 and give some examples and counterexamples that can be realized by a ladder Ore extension.

Infinite dimensional serial algebras and their representations

We study arbitrary algebras all of whose indecomposable finite dimensional representations are uniserial, or more generally, comodules over serial coalgebras. We show that the infinite representation theory for such algebras parallels and non-trivially extends the theory of primary infinite abelian groups: we obtain a Kulikov and a Prüfer decomposability criterion and determine indecomposables in the category of locally finite modules over such an algebra. As applications, we obtain consequences on the infinite representation theory of the extended affine Dynkin quiver An˜, and the infinite line and half line quivers, and obtain Jordan form results for operators on infinite dimensional spaces. We also find a class of counterexamples to an infinite version of the pure semisimplicity conjecture: there exist algebras all of whose locally finite left modules are direct sums of indecomposables, but not all of whose right locally finite modules decompose as such. On the way, we also provide classifications of one-sided serial (co)algebras, and introduce and classify a class of quantum groups whose indecomposable representations are uniserial, namely, coserial pointed Hopf algebras. These are one-parameter deformations of group algebras and are obtained as certain Ore extensions of these.

Discriminants of Taft Algebra Smash Products and Applications

A general criterion is given for when the center of a Taft algebra smash product is the fixed ring. This is applied to the study of the noncommutative discriminant. Our method relies on the Poisson methods of Nguyen, Trampel, and Yakimov, but also makes use of Poisson Ore extensions. Specifically, we fully determine the inner faithful actions of Taft algebras on quantum planes and quantum Weyl algebras. We compute the discriminant of the corresponding smash product and apply it to compute the Azumaya locus and restricted automorphism group.

D-groups and the Dixmier–Moeglin equivalence

A differential-algebraic geometric analogue of the Dixmier-Moeglin equivalence is articulated, and proven to hold for $D$-groups over the constants. The model theory of differentially closed fields of characteristic zero, in particular the notion of analysability in the constants, plays a central role. As an application it is shown that if $R$ is a commutative affine Hopf algebra over a field of characteristic zero, and $A$ is an Ore extension to which the Hopf algebra structure extends, then $A$ satisfies the classical Dixmier-Moeglin equivalence. Along the way it is shown that all such $A$ are Hopf Ore extensions

Generalized Hopf–Ore extensions

We investigate some Hopf algebra structures over an Ore extension of a Hopf algebra. Necessary and sufficient conditions for an Ore extension of a Hopf algebra to have a Hopf algebra structure of a certain type are given. This construction, called a generalized Hopf–Ore extension, generalizes Hopf–Ore extensions. We describe the generalized Hopf–Ore extensions of the enveloping algebras of Lie algebras. For some Lie algebras g, the generalized Hopf–Ore extensions of U(g) are classified.

Enveloping algebras of double Poisson-Ore extensions

ABSTRACTIt is proved that the Poisson enveloping algebra of a double Poisson-Ore extension is an iterated double Ore extension. As an application, properties that are preserved under iterated double Ore extensions are invariants of the Poisson enveloping algebra of a double Poisson-Ore extension.

Gelfand–Kirillov dimension of generalized Weyl algebras (In memory of Guenter Rudolf Krause (1941–2015))

ABSTRACTGiven a generalized Weyl algebra A of degree 1 with the base algebra D, we prove that the difference of the Gelfand–Kirillov dimension of A and that of D could be any positive integer or infinity. Under mild conditions, this difference is exactly 1. As applications, we calculate the Gelfand–Kirillov dimensions of various algebras of interest, including the (quantized) Weyl algebras, ambiskew polynomial rings, noetherian (generalized) down-up algebras, iterated Ore extensions, quantum Heisenberg algebras, universal enveloping algebras of Lie algebras, quantum GWAs, etc.

Non-associative Ore extensions

We introduce non-associative Ore extensions, S = R[X; σ, δ], for any nonassociative unital ring R and any additive maps σ, δ: R → R satisfying σ(1) = 1 and δ(1) = 0. In the special case when δ is either left or right R δ -linear, where R δ = ker(δ), and R is δ-simple, i.e. {0} and R are the only δ-invariant ideals of R, we determine the ideal structure of the nonassociative differential polynomial ring D = R[X; id R , δ]. Namely, in that case, we show that all non-zero ideals of D are generated by monic polynomials in the center Z(D) of D. We also show that Z( D ) = R δ [p] for a monic p ∈ R δ [X], unique up to addition of elements from Z(R) δ . Thereby, we generalize classical results by Amitsur on differential polynomial rings defined by derivations on associative and simple rings. Furthermore, we use the ideal structure of D to show that D is simple if and only if R is δ-simple and Z(D) equals the field R δ ∩ Z(R). This provides us with a non-associative generalization of a result by Oinert, Richter and Silvestrov. This result is in turn used to show a non-associative version of a classical result by Jordan concerning simplicity of D in the cases when the characteristic of the field R δ ∩ Z(R) is either zero or a prime. We use our findings to show simplicity results for both non-associative versions of Weyl algebras and non-associative differential polynomial rings defined by monoid/group actions on compact Hausdorff spaces.

On the ACCP in skew Poincaré–Birkhoff–Witt extensions

The objective of this paper is to investigate the ascending chain condition on principal left (resp. right) ideals of noncommutative rings known as skew PBW extensions. We consider these extensions over domains and over \(\varSigma \)-rigid rings. We unify and extend several results in the literature for Ore extensions and another families of rings which can not be expressed as iterated Ore extensions.

Hom-associative Ore extensions

We introduce hom-associative Ore extensions as non-associative, non-unital Ore extensions with a hom-associative multiplication, as well as give some necessary and sufficient conditions when such exist. Within this framework, we also construct a family of hom-associative Weyl algebras as generalizations of the classical analogue, and prove that they are simple.

\sigma-PBW EXTENSIONS OF SKEW \Pi-ARMENDARIZ RINGS

In this paper we present the notion of skew $\Pi$-Armendariz for the non-commutative rings known as $\sigma$-PBW extensions. This concept generalizes several definitions of Armendariz rings presented in the literature for these extensions, and in particular, for Ore extensions of injective type. We investigate the relations between skew $\Pi$-Armendariz, $\Sigma$-rigid, $(\Sigma,\Delta)$-skew Armendariz and $\Sigma$-skew Armendariz rings rings.

Actions of Ore extensions and growth of polynomial H-identities

ABSTRACTWe show that if A is a finite-dimensional associative H-module algebra for an arbitrary Hopf algebra H, then the proof of the analog of Amitsur’s conjecture for H-codimensions of A can be reduced to the case when A is H-simple. (Here we do not require that the Jacobson radical of A is an H-submodule.) As an application, we prove that if A is a finite-dimensional associative H-module algebra where H is a Hopf algebra H over a field of characteristic 0 such that H is constructed by an iterated Ore extension of a finite-dimensional semisimple Hopf algebra by skew-primitive elements (e.g., H is a Taft algebra), then there exists integer PIexpH(A). In order to prove this, we study the structure of algebras simple with respect to an action of an Ore extension.

A notion of compatibility for Armendariz and Baer properties over skew PBW extensions

In this paper we are interested in studying the properties of Armendariz, Baer, quasi-Baer, p.p. and p.q.-Baer over skew PBW extensions. Using a notion of compatibility, we generalize several propositions established for Ore extensions and present new results for several noncommutative rings which can not be expressed as Ore extensions (universal enveloping algebras, diffusion algebras, and others).

Some types of ring elements in Ore extensions over noncommutative rings

Let [Formula: see text] be an associative unital ring with an endomorphism [Formula: see text] and [Formula: see text]-derivation [Formula: see text]. Some types of ring elements such as the units and the idempotents play distinguished roles in noncommutative ring theory, and will play a central role in this work. In fact, we are interested to study the unit elements, the idempotent elements, the von Neumann regular elements, the [Formula: see text]-regular elements and also the von Neumann local elements of the Ore extension ring [Formula: see text], when the base ring [Formula: see text] is a right duo ring which is [Formula: see text]-compatible. As an application, we completely characterize the clean elements of the Ore extension ring [Formula: see text], when the base ring [Formula: see text] is a right duo ring which is [Formula: see text]-compatible.